I want to state first that I do not understand the Physics of this situation. I have tried to "reverse engineer" the problem and determine spring constants, initial positions etc, from the x(t) equation and the initial conditions and have come up with conflicting facts. Perhaps there is just something here I'm not seeing, but I would double check to make sure your general solution here is valid.

I see what has happened here. You didn't list all of your iniitial conditions. You needed to state the condition:

So what we have here is a general solution

with
and

For convenience, I'm going to define
and

So your problem becomes:
with
and

Let's work with the velocity condition first:

For t = 0:

Thus . Which one we choose is going to depend on which spring is exerting the greater force. You have not given us that information. I will show you how the equation shows this factor.

So the solution will take the form of: or
with

Now,

Thus the solution is
with
Which one of the + or - depends on the phase angle.

So to continue:

The size of L is going to determine if A is positive or negative. I leave it to you to decide if can be negative or not. (The general solution for x(t) seems to imply one thing and the initial condition seems to imply another so I cannot reliably predict the size of L.)

I will simply leave this as

As a curious consequence of the initial condition notice that
The signs are taken to be in the same sense, so they cancel out, leaving
which, when put into the form of the original variables leads to

(We also see that I was right: You didn't use parenthesis in your original post, as this is obviously what you had meant to write.)

As to the rest, yes, the amplitude is indeed

But has nothing at all to do with dx/dt = v. I don't know why you would say that. This is not true, if for no other line of reasoning, because the units are wrong. And besides
in general as you can see by taking the derivative of x(t).